Problem 14 - Precalculus

Question

Express each complex number in polar form. $$\sqrt{5}-\sqrt{5} i$$

Step-by-Step Solution

Answer

The polar form is \( \sqrt{10}e^{-i\frac{\pi}{4}} \).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \( \sqrt{5} - \sqrt{5}i \). Here, the real part is \( \sqrt{5} \) and the imaginary part is \( - \sqrt{5} \).

Step 2: Calculate the Magnitude

The magnitude \( r \) of a complex number \( a + bi \) is found using the formula \( r = \sqrt{a^2 + b^2} \). For \( \sqrt{5} - \sqrt{5}i \), the magnitude is \( r = \sqrt{(\sqrt{5})^2 + (-\sqrt{5})^2} = \sqrt{5 + 5} = \sqrt{10} \).

Step 3: Calculate the Argument

The argument \( \theta \) is calculated using \( \tan \theta = \frac{b}{a} \). Here, \( a = \sqrt{5} \) and \( b = -\sqrt{5} \), so \( \tan \theta = \frac{-\sqrt{5}}{\sqrt{5}} = -1 \). The angle \( \theta \) for which \( \tan \theta = -1 \) is \( -\frac{\pi}{4} \) (or equivalently \( \frac{7\pi}{4} \)) on the unit circle.

Step 4: Express in Polar Form

The polar form of a complex number is given by \( r (\cos \theta + i \sin \theta) \) or \( r e^{i\theta} \). Thus, \( \sqrt{5} - \sqrt{5}i \) in polar form is \( \sqrt{10} (\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})) \). This can also be written as \( \sqrt{10}e^{-i\frac{\pi}{4}} \).

Key Concepts

Magnitude of Complex Numbers

The magnitude of a complex number represents its distance from the origin in the complex plane. To find this magnitude, we use the formula:

• Magnitude, denoted as r, can be found using: \( r = \sqrt{a^2 + b^2} \)

where \( a \) is the real part and \( b \) is the imaginary part of the complex number.
In our exercise, the complex number is \( \sqrt{5} - \sqrt{5}i \).

• The real part \( a \) is \( \sqrt{5} \).
• The imaginary part \( b \) is \( -\sqrt{5} \).

The magnitude is calculated as \( \sqrt{(\sqrt{5})^2 + (-\sqrt{5})^2} \).
Simplifying, this becomes \( \sqrt{5 + 5} = \sqrt{10} \).This tells us that our complex number, \( \sqrt{5} - \sqrt{5}i \), is \( \sqrt{10} \) units away from the origin.

Argument of Complex Numbers

The argument of a complex number is essentially the angle that the line representing the number makes with the positive real axis, in the complex plane.
Calculating this angle, known as \( \theta \), involves:

• Using the formula: \( \tan \theta = \frac{b}{a} \)

For the given complex number \( \sqrt{5} - \sqrt{5}i \),

• \( a = \sqrt{5} \)
• \( b = -\sqrt{5} \)

Substitute these values in: \( \tan \theta = \frac{-\sqrt{5}}{\sqrt{5}} = -1 \).
The angle \( \theta \) for which \( \tan \theta = -1 \) occurs at \( -\frac{\pi}{4} \), or \( \frac{7\pi}{4} \) in positive angle notation.
This gives us the direction in which our complex number is positioned relative to the positive x-axis.

Conversion to Polar Form

Converting a complex number into its polar form involves using both its magnitude and argument.
This representation allows us to express the complex number in terms of radius and angle, making calculations like multiplication and division more intuitive.
The polar form is given by:

• \( r(\cos \theta + i\sin \theta) \)
• Or equivalently, \( re^{i\theta} \)

For our example, \( \sqrt{5} - \sqrt{5}i \), we've already calculated:

• The magnitude \( \sqrt{10} \)
• The argument \( -\frac{\pi}{4} \)

Thus, the polar form of \( \sqrt{5} - \sqrt{5}i \) is expressed as \( \sqrt{10} (\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4})) \).
Alternatively, using exponential form, it can be written as \( \sqrt{10}e^{-i\frac{\pi}{4}} \).
The polar form captures not just the magnitude but also the orientation of the complex number on the complex plane.